INDETERMINATE ANALYSIS. 



and thus the reduced equation becomes 



u' — 1001'= — 1302S4. 

 In which, having found « =346, and i = 53, 

 we have C, == 'UZl = MLzJ^ = ,, 



1 - t-by-'i _ 5£^li3 jii. _ , 

 V ~ ia ~ 6 ~ ^^ 



therefore .r = 4, and ^ = 3, arc the values of x and_y in the 

 original equation. 



It will be obferved that we have employed here the molt 

 general form that equations of this kind admit of, and 

 therefore the formulae ai-e more complex than they ufually 

 occur in practical cafes, for when any of the co-efficients ,j, 

 b, c, Sec. become zero, the expreffions are much limplified, 

 as appears from the following example. 



Ex. 2. — Reduce 7.1-'- 4- ^.ly + y'- = 67 to its fimpleft 

 form. 



Here a = 7, b = §, c = i, </ = o, ? = o, / = — 67, 

 whence, omitting thofe quantities that are zero, we have 

 ri' — 4 « /: = A = 29; and^r^ o 

 J^^ - 4«/= /} = 1876 



Again, if A > B, then tlie remainder arifing from -- will 



be found in the lower feries, and the number B in the upper 

 fciies of remainders, if the equation be poffible ; and u 

 tliefe conditions have not place, the propofed equation is iiu- 

 poffible. 



NoU. — It is to be obferved, that equations falling und'-r 

 the poffible form are not aUvays folvible in integers, tlif 

 proof extending only to their folvibility in rational numbc; 

 which may therefore lometimes be fraftional ; but when 1 1 

 fail under animpoffible form, tliey will admit of no folutii 

 either in integers or fraftions. 



E.v. —Required the poffibility or impoffibihty of the cfji 

 tion 



. remamder . 



29. 1S76' 



B 



and thus the reduced equation is 



a- — 29 /" = 29 . 1S76-. 



And in a fimilar manner may any indeterminate equation 

 of the fecond degree be reduced to the form «■ — A /" =: B. 



Having therefore ffieMrn the method of reducing every in- 

 determinate equation to the form ir — A t" =:B ; it follows 

 that thefolution of this fimplo form involves with it the fo- 

 lution of every equation of this kind that can be propofed ; 

 we (liall therefore, in the following propofition, attend to the 

 folution of this particular cafe. But it may be proper to 

 ftate, that there are an infinite number of equations of this 

 kind that are impoffible ; and will admit of no folution, either 

 in integers or fraSions ; and tlierefore before we proceed 

 farther in the invelligation, it will be afeful to lay down a 

 rule, whence their poITibility, or impoffibihty, may be af- 

 certained ; as we may thus frequently fave much unneceflary 

 labour. 



Prop. XIII. 

 To afcertain the poffibihty or impoffibihty of every in- 

 determinate equation of the fecond degree. 



Rule, — Reduce the propofed equation to the form 

 u' - At'= B, 

 and find all the remainders arifing from dividing each of tlie 



fquares i, 2', 3', 4% &c. ( j by A ; 



and alfo the remainders arifing from dividing each of the 



fquares i% 2', 3', 4% Sic. (■ j by B ; 



and again, divide the greatefl of thefe numbers A, B, by 

 thelcallof them, and obferve the remainder. 



Then if B be greater than A, this laft remainder will be 

 found amongft thofe of the upper feries ; and the number A 

 will be found amongll thofe of the lower feries, if the equa- 

 tion be poffible. 



And convcrfcly, if thefe conditions have not place, the 

 propofed equation will admit of no folution, either in in- 

 tegers or fraAions. 



Now 4 is found in the upper feries of remainders, but 7 is 

 not found in the lower ; therefore the equation cannot have 

 place either in integers or fraftions. 



And for the fame reafon, the equation x' — J y' = II z' 

 is alio impoffible, for if tliis was poffible, fo would like- 



; I ; which we have feen is impof- 



wif^ ^ - ; l,_ 

 fible. " 



Ex. 2. — It is reqiiired to afcertain the poffibility or itn- 

 poffibility of the equation 



fquares i% 2', 3", 4', 5", 6' divided by 13 



remainders i, 4, 9, 3, 12, 10 

 And the fame fquares, divided by 12, give for 

 remainders i, 4, 9> 4> i> O 



alfo, — = I and remainder i. 

 12 



And here, fince 12 is found in th« upper feries, and I l.-i 

 the lower, the equation is folvible. 



Note. — If tlie equation propofed be of the form 



x---Ay-= -B, 



we muft employ, inftead of the pofitive remainders ariil 

 from A, the negative remainders of the fame, that is, tak'. 

 the quotients in excefs. And if the equation have the fi>r 



X- + A/ = B, 

 then we mull employ the negative remainders of B. Hav- 

 ing thus given an idea of the method of judging of the pof- 

 fibility of every equation of the form .v' — Ay' = B ; and 

 having alfo Ihewn how any indeterminate of the iccond degree 

 may be reduced to this form, it only remains to {hew the 

 method of folution of the above equation ; or, which is Hill 

 a more general form, of the equation 



but in this, as in the other propofitioiis, we can only indicate 

 the method, without attempting to inveftigate the rationale 

 of the operation ; as this would carry us much beyond our 

 limits. Now it is fliewn under the article Diophaniine, tha^ 



the 



